I had been recently watching a video and had found out that by taking the length of any A size paper and dividing it by its width, you got exactly \(\sqrt {2}\). The story behind this was simple. Before, the government had wanted to created a standard piece of paper that, when cut through its length in half, when you divide the length by the width, you would still have the same answer.

This is how it was figured out:

\(\frac {a}{b} = \frac {b}{\frac {1}{2} a}\)

\(2b^2 = a^2\)

\(\sqrt {2} b = a\)

\(\frac {a}{b} = \sqrt {2}\)

Where else do you think \(\sqrt {2}\) is used in our modern day life?

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